Graph f(x)=x^2-3x-2

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Trigonometry

Graph f(x)=x^2-3x-2

$f (x) = x^{2} - 3 x - 2$

Find the properties of the given parabola.

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Rewrite the equation in vertex form.

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Complete the square for $x^{2} - 3 x - 2$ .

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Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1, b = - 3, c = - 2$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = - \frac{3}{2 (1)}$

Multiply $2$ by $1$ .

$d = - \frac{3}{2}$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Simplify each term.

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Raise $- 3$ to the power of $2$ .

$e = - 2 - \frac{9}{4 (1)}$

Multiply $4$ by $1$ .

$e = - 2 - \frac{9}{4}$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 2 \cdot \frac{4}{4} - \frac{9}{4}$

Combine $- 2$ and $\frac{4}{4}$ .

$e = \frac{- 2 \cdot 4}{4} - \frac{9}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 2 \cdot 4 - 9}{4}$

Simplify the numerator.

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Multiply $- 2$ by $4$ .

$e = \frac{- 8 - 9}{4}$

Subtract $9$ from $- 8$ .

$e = \frac{- 17}{4}$

Move the negative in front of the fraction.

$e = - \frac{17}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $a {(x + d)}^{2} + e$ .

${(x - \frac{3}{2})}^{2} - \frac{17}{4}$

Set $y$ equal to the new right side.

$y = {(x - \frac{3}{2})}^{2} - \frac{17}{4}$

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 1$

$h = \frac{3}{2}$

$k = - \frac{17}{4}$

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Find the vertex $(h, k)$ .

$(\frac{3}{2}, - \frac{17}{4})$

Find $p$ , the distance from the vertex to the focus.

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Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 1}$

Cancel the common factor of $1$ .

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Cancel the common factor.

$\frac{1}{4 \cdot 1}$

Rewrite the expression.

$\frac{1}{4}$

Find the focus.

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The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(\frac{3}{2}, - 4)$

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = \frac{3}{2}$

Find the directrix.

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The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = - \frac{9}{2}$

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{17}{4})$

Focus: $(\frac{3}{2}, - 4)$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{9}{2}$

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{17}{4})$

Focus: $(\frac{3}{2}, - 4)$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{9}{2}$

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Replace the variable $x$ with $1$ in the expression.

$f (1) = {(1)}^{2} - 3 \cdot 1 - 2$

Simplify the result.

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Simplify each term.

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One to any power is one.

$f (1) = 1 - 3 \cdot 1 - 2$

Multiply $- 3$ by $1$ .

$f (1) = 1 - 3 - 2$

Simplify by subtracting numbers.

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Subtract $3$ from $1$ .

$f (1) = - 2 - 2$

Subtract $2$ from $- 2$ .

$f (1) = - 4$

The final answer is $- 4$ .

$- 4$

The $y$ value at $x = 1$ is $- 4$ .

$y = - 4$

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{2} - 3 \cdot 0 - 2$

Simplify the result.

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Simplify each term.

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Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - 3 \cdot 0 - 2$

Multiply $- 3$ by $0$ .

$f (0) = 0 + 0 - 2$

Simplify by adding and subtracting.

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Add $0$ and $0$ .

$f (0) = 0 - 2$

Subtract $2$ from $0$ .

$f (0) = - 2$

The final answer is $- 2$ .

$- 2$

The $y$ value at $x = 0$ is $- 2$ .

$y = - 2$

Replace the variable $x$ with $3$ in the expression.

$f (3) = {(3)}^{2} - 3 \cdot 3 - 2$

Simplify the result.

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Simplify each term.

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Raise $3$ to the power of $2$ .

$f (3) = 9 - 3 \cdot 3 - 2$

Multiply $- 3$ by $3$ .

$f (3) = 9 - 9 - 2$

Simplify by subtracting numbers.

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Subtract $9$ from $9$ .

$f (3) = 0 - 2$

Subtract $2$ from $0$ .

$f (3) = - 2$

The final answer is $- 2$ .

$- 2$

The $y$ value at $x = 3$ is $- 2$ .

$y = - 2$

Replace the variable $x$ with $4$ in the expression.

$f (4) = {(4)}^{2} - 3 \cdot 4 - 2$

Simplify the result.

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Simplify each term.

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Raise $4$ to the power of $2$ .

$f (4) = 16 - 3 \cdot 4 - 2$

Multiply $- 3$ by $4$ .

$f (4) = 16 - 12 - 2$

Simplify by subtracting numbers.

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Subtract $12$ from $16$ .

$f (4) = 4 - 2$

Subtract $2$ from $4$ .

$f (4) = 2$

The final answer is $2$ .

$2$

The $y$ value at $x = 4$ is $2$ .

$y = 2$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - 2 \\ 1 & - 4 \\ \frac{3}{2} & - \frac{17}{4} \\ 3 & - 2 \\ 4 & 2 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{17}{4})$

Focus: $(\frac{3}{2}, - 4)$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{9}{2}$

$\begin{array}{c} x & y \\ 0 & - 2 \\ 1 & - 4 \\ \frac{3}{2} & - \frac{17}{4} \\ 3 & - 2 \\ 4 & 2 \end{array}$

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Name

Name	seven hundred sixty-six million four hundred thirty thousand four hundred four

Interesting facts

766430404 has 16 divisors, whose sum is 1725221268
The reverse of 766430404 is 404034667
Previous prime number is 2357

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 34
Digital Root 7

Name

Name	one billion three hundred forty-six million three hundred twenty-six thousand six hundred sixty-eight

Interesting facts

1346326668 has 32 divisors, whose sum is 3371029920
The reverse of 1346326668 is 8666236431
Previous prime number is 653

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 45
Digital Root 9

Name

Name	one billion eight hundred sixty-three million one hundred thirty-seven thousand three hundred fourteen

Interesting facts

1863137314 has 8 divisors, whose sum is 2795724288
The reverse of 1863137314 is 4137313681
Previous prime number is 2767

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 37
Digital Root 1

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